Our theory both simplifies and strengthens existing analyses in online confor-mal prediction.

We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of this ODE using standard Runge-Kutta integrators. Specifically, we prove that under Lipschitz-gradient, convexity and order-$(s+2)$ differentiability assumptions, the sequence of iterates generated by discretizing the proposed second-order ODE converges to the optimal solution at a rate of $\mathcal{O}({N^{-2\frac{s}{s+1}}})$, where $s$ is the order of the Runge-Kutta numerical integrator. Furthermore, we introduce a new local flatness condition on the objective, under which rates even faster than $\mathcal{O}(N^{-2})$ can be achieved with low-order integrators and only gradient information. Notably, this flatness condition is satisfied by several standard loss functions used in machine learning. We provide numerical experiments that verify the theoretical rates predicted by our results.

This paper is concerned with differentiable resampling in the context of sequential Monte Carlo (e.g., particle filtering). We propose a new informative resampling method that is instantly pathwise differentiable, based on an ensemble score diffusion model. We prove that our diffusion resampling method provides a consistent estimate to the resampling distribution, and we show by experiments that it outperforms the state-of-the-art differentiable resampling methods when used for stochastic filtering and parameter estimation.

The rapid expansion of platform integration has emerged as an effective solution to mitigate market fragmentation by consolidating multiple ride-hailing platforms into a single application. To address heterogeneous passenger preferences, third-party integrators provide Discount Express service delivered by express drivers at lower trip fares. For the individual platform, encouraging broader participation of drivers in Discount Express services has the potential to expand the accessible demand pool and improve matching efficiency, but often at the cost of reduced profit margins. This study aims to dynamically manage drivers' acceptance of Discount Express from the perspective of an individual platform. The lack of historical data under the new business model necessitates online learning. However, early-stage exploration through trial and error can be costly in practice, highlighting the need for reliable early-stage performance in real-world deployment. To address these challenges, this study formulates the decision regarding the proportion of drivers accepting discount orders as a continuous control task. In response to the high stochasticity and the opaque matching mechanisms employed by third-party integrator, we propose an innovative policy-improved deep deterministic policy gradient (pi-DDPG) framework. The proposed framework incorporates a refiner module to boost policy performance during the early training phase. A customized simulator based on a real-world dataset is developed to validate the effectiveness of the proposed pi-DDPG. Numerical experiments demonstrate that pi-DDPG achieves superior learning efficiency and significantly reduces early-stage training losses, enhancing its applicability to practical ride-hailing scenarios.

We propose two stochastic gradient MCMC methods for sampling from Bayesian posterior distributions defined on Riemann manifolds with a known geodesic flow, e.g.

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Dynamics-based Markov Chain Monte Carlo methods (D-MCMCs) are sampling methods using dynamics simulation for state transition in a Markov chain.